Triangle inequalities are really useful for figuring out what kind of triangle we have, especially in our Grade 12 geometry class.

Let's talk about what the triangle inequality theorem says. It tells us that if we take any two sides of a triangle, their lengths added together must be greater than the length of the third side. This helps us understand how the sides of the triangle relate to each other.

How Triangle Inequalities Work:

1. Basic Rule: For any triangle with sides labeled as $a$, $b$, and $c$, these rules must be true:
   • $a + b > c$
   • $a + c > b$
   • $b + c > a$

If all these rules hold, then we definitely have a triangle!

2. Finding Types of Triangles:

   • By Sides:

     • Equilateral: All three sides are the same length ($a = b = c$). The triangle inequalities work here since the sums will equal each side.
     • Isosceles: Two sides are the same length (for example, $a = b$). The inequalities still work because $2a > c$ ensures that we have a triangle.
     • Scalene: All sides are different lengths ($a \neq b \neq c$). In this case, the inequalities help us make sure that those sides can actually form a triangle.

   • By Angles:

     • We can also figure out the angles from the sides. For example, if $a^2 + b^2 < c^2$, we can tell that we have an obtuse triangle (one angle is greater than 90 degrees). If $a^2 + b^2 = c^2$, then it’s a right triangle (one angle is exactly 90 degrees). And if $a^2 + b^2 > c^2$, we have an acute triangle (all angles are less than 90 degrees).

Conclusion:

So, triangle inequalities help us not only see if a triangle can exist but also tell us what type of triangle it is based on the lengths of its sides. It’s interesting to see how these simple rules give us so much information. It feels like solving a puzzle where each piece—the sides and angles—fits together perfectly!